## Optimal Coding and Sampling of Triangulations (2003)

### Cached

### Download Links

- [www.liafa.jussieu.fr]
- [www.lix.polytechnique.fr]
- DBLP

### Other Repositories/Bibliography

Citations: | 36 - 5 self |

### BibTeX

@INPROCEEDINGS{Poulalhon03optimalcoding,

author = {Dominique Poulalhon and Gilles Schaeffer},

title = {Optimal Coding and Sampling of Triangulations},

booktitle = {},

year = {2003},

pages = {1080--1094},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1

### Citations

275 | Edgebreaker: Connectivity compression for triangle meshes
- ROSSIGNAC
- 1999
(Show Context)
Citation Context ... very effective “structure driven” approach consists in distinguishing the encoding of the combinatorial structure, – that is, the triangulation – from the geometry – that is, vertex coordinates (see =-=[26]-=- for a survey and [16] for an opposite “coordinate driven” approach). Three main properties of the combinatorial code are then desirable: compacity, that is minimization of the bit length of code word... |

198 |
Embedding planar graphs on the grid
- Schnyder
- 1990
(Show Context)
Citation Context ...���� � � �� � � � �� � � ��� � � � � � �� T4 = 13 � Fig. 2. The smallest triangulations with their inequivalent rootings. Fig. 3. The 9 elements of the set B3. positions, where introduced by Schnyder =-=[29]-=- to compute graph embeddings and have proved a fundamental tool in the study of planar graphs [8, 10, 14, 20]. The role played in this paper by minimal realizers of triangulations is akin to the role ... |

118 |
A census of planar triangulations
- Tutte
- 1962
(Show Context)
Citation Context ...ramework: known methods to achieve locality require the code to be based on a spanning tree of the graph. Counting. The exact enumeration problem for triangulations was solved by Tutte in the sixties =-=[30]-=-. The number of rooted triangulations with 2n triangles, 3n edges and n + 2 vertices is Tn = (This formula gives the previous constant α0 = 256 27 2 (4n − 3)! . (1) n!(3n − 1)! .) More generally Tutte... |

114 |
Cutsem. A calculus for the random generation of labelled combinatorial structures
- Flajolet, Zimmermann, et al.
- 1994
(Show Context)
Citation Context ...plers of at least quadratic complexities. On the other hand, algorithms of the second type take advantage of exact counting results to construct directly a configuration from the uniform distribution =-=[15]-=-. As a result these perfect samplers often operate in linear time with little more than the amount of random bits required by information theory bounds to generate a configuration [2, 13]. It is very ... |

68 | Progressive lossless compression of arbitrary simplicial complexes
- Gandoin, Devillers
- 2002
(Show Context)
Citation Context ...ture driven” approach consists in distinguishing the encoding of the combinatorial structure, – that is, the triangulation – from the geometry – that is, vertex coordinates (see [26] for a survey and =-=[16]-=- for an opposite “coordinate driven” approach). Three main properties of the combinatorial code are then desirable: compacity, that is minimization of the bit length of code words, linear complexity o... |

67 | Drawing planar graphs using the canonical ordering
- Kant
- 1996
(Show Context)
Citation Context ... inequivalent rootings. Fig. 3. The 9 elements of the set B3. positions, where introduced by Schnyder [29] to compute graph embeddings and have proved a fundamental tool in the study of planar graphs =-=[8, 10, 14, 20]-=-. The role played in this paper by minimal realizers of triangulations is akin to the role of breadth-first search spanning trees in planar maps [28], and of minimal bipolar orientations in 2-connecte... |

59 | Guaranteed 3.67V bit encoding of planar triangle graphs
- King, Rossignac
(Show Context)
Citation Context ...fundamental class Tn of triangulations of a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in [6, 11, 18], to α = 3.67 in =-=[21, 28]-=-, and recently α = 3.37 bits in [7]. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an optimal solution for general ... |

55 |
Quantum Geometry
- Ambjørn, Durhuus, et al.
- 1997
(Show Context)
Citation Context ...that the propersFig. 1. A random triangulation with 30 triangles. discretization of a typical quantum universe is precisely obtained by sampling from the uniform distribution on rooted triangulations =-=[4]-=-. Several approximate sampling algorithms were thus developed by physicists for planar maps, including for triangulations [3]. Most of them are based on Markov chains, the mixing times of which are no... |

46 | Compact encodings of planar graphs via canonical orderings and multiple parentheses
- Chuang, Garg, et al.
(Show Context)
Citation Context ... small increments). For the fundamental class Tn of triangulations of a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in =-=[6, 11, 18]-=-, to α = 3.67 in [21, 28], and recently α = 3.37 bits in [7]. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an opti... |

43 | Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees
- Schaeffer
- 1997
(Show Context)
Citation Context ...atorial proofs explaining these results, as opposed to the technical computational proofs à la Tutte. This was done in a very general setting for maps without restrictions on multiple edges and loops =-=[9, 27]-=-. However these methods do not apply to the case of triangulations. It should be stressed that planar graphs have in general non-unique embeddings: a given planar graph may underlie many planar maps. ... |

31 | Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes. Order 18:19–37
- Felsner
- 2001
(Show Context)
Citation Context ... inequivalent rootings. Fig. 3. The 9 elements of the set B3. positions, where introduced by Schnyder [29] to compute graph embeddings and have proved a fundamental tool in the study of planar graphs =-=[8, 10, 14, 20]-=-. The role played in this paper by minimal realizers of triangulations is akin to the role of breadth-first search spanning trees in planar maps [28], and of minimal bipolar orientations in 2-connecte... |

29 |
Random sampling of large planar maps and convex polyhedra
- Schaeffer
- 1999
(Show Context)
Citation Context ...fundamental class Tn of triangulations of a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in [6, 11, 18], to α = 3.67 in =-=[21, 28]-=-, and recently α = 3.37 bits in [7]. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an optimal solution for general ... |

27 |
The degree distribution in bipartite planar maps: applications to the Ising model
- Bousquet-Melou, Schaeffer
- 2002
(Show Context)
Citation Context ...atorial proofs explaining these results, as opposed to the technical computational proofs à la Tutte. This was done in a very general setting for maps without restrictions on multiple edges and loops =-=[9, 27]-=-. However these methods do not apply to the case of triangulations. It should be stressed that planar graphs have in general non-unique embeddings: a given planar graph may underlie many planar maps. ... |

25 | On random planar graphs, the number of planar graphs and their triangulations
- Osthus, Prömel, et al.
(Show Context)
Citation Context ...y planar maps. This explains that, as opposed to the situation for maps, no exact formula is known for the number of planar graphs with n vertices (even the asymptotic growth factor is not known, see =-=[7, 23]-=-). However according to Whitney’s theorem, 3-connected planar graphs have an essentially unique embedding. In particular the class of triangulations is equivalent to the class of maximal planar graphs... |

24 |
Almost all maps are asymmetric
- Richmond, Wormald
- 1995
(Show Context)
Citation Context ...robability to output a specific rooted triangulation T with 2n vertices is (resp. is close to) 1/Tn. Safe for an exponentially small fraction of them, triangulations have a trivial automorphism group =-=[25]-=-, so that as far as polynomial parameters are concerned, the uniform distribution on rooted or unrooted triangulations are indistinguishable. This question was first considered by physicists willing t... |

23 | A fast general methodology for information-theoretically optimal encodings of graphs
- He, Kao, et al.
(Show Context)
Citation Context ...mation theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an optimal solution for general recursive classes of planar maps by Lu et al. =-=[19, 22]-=-. For a fixed ⋆ Extended abstract submitted to ICALP2003, Track A.s2 class, say triangulations, this algorithm does not use the knowledge of α0, as expected for a generic algorithm, and instead relies... |

21 | An information-theoretic upper bound of planar graphs using triangulation
- Bonichon, Gavoille, et al.
- 2003
(Show Context)
Citation Context ...f a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in [6, 11, 18], to α = 3.67 in [21, 28], and recently α = 3.37 bits in =-=[7]-=-. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an optimal solution for general recursive classes of planar maps by... |

21 |
Annotated bibliography of perfectly random sampling with Markov chains. Microsurveys in discrete probability
- Wilson
- 1997
(Show Context)
Citation Context ...til it has (approximately) forgotten its start point. This is a very versatile method that requires little knowledge of the structures. It can even allow for perfect sampling in some restricted cases =-=[31]-=-. However in most cases it yields only approximate samplers of at least quadratic complexities. On the other hand, algorithms of the second type take advantage of exact counting results to construct d... |

20 | Linear-time succinct encodings of planar graphs via canonical orderings
- He, Kao, et al.
- 1999
(Show Context)
Citation Context ... small increments). For the fundamental class Tn of triangulations of a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in =-=[6, 11, 18]-=-, to α = 3.67 in [21, 28], and recently α = 3.37 bits in [7]. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an opti... |

18 |
A bijection between realizers of maximal plane graphs and pairs of noncrossing Dyck paths
- Bonichon
- 2005
(Show Context)
Citation Context ... small increments). For the fundamental class Tn of triangulations of a sphere with 2n triangles, several codes of linear complexity were proposed, with various bit length αn(1 + o(1)): from α = 4 in =-=[6, 11, 18]-=-, to α = 3.67 in [21, 28], and recently α = 3.37 bits in [7]. The information theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an opti... |

16 |
3-orientations and Schnyder 3-tree-decompositions. Diplomarbeit, Freie Universität
- Brehm
- 2000
(Show Context)
Citation Context ... inequivalent rootings. Fig. 3. The 9 elements of the set B3. positions, where introduced by Schnyder [29] to compute graph embeddings and have proved a fundamental tool in the study of planar graphs =-=[8, 10, 14, 20]-=-. The role played in this paper by minimal realizers of triangulations is akin to the role of breadth-first search spanning trees in planar maps [28], and of minimal bipolar orientations in 2-connecte... |

14 | Linear-time compression of bounded-genus graphs into information-theoretically optimal number of bits
- Lu
- 2002
(Show Context)
Citation Context ...mation theory bound on α is α0 = 1 n log |Tn| ∼ 256 27 ≈ 3.245 (see below). In some sense the compacity problem was given an optimal solution for general recursive classes of planar maps by Lu et al. =-=[19, 22]-=-. For a fixed ⋆ Extended abstract submitted to ICALP2003, Track A.s2 class, say triangulations, this algorithm does not use the knowledge of α0, as expected for a generic algorithm, and instead relies... |

13 | Random sampling from Boltzmann principles
- Duchon, Flajolet, et al.
(Show Context)
Citation Context ...rm distribution [15]. As a result these perfect samplers often operate in linear time with little more than the amount of random bits required by information theory bounds to generate a configuration =-=[2, 13]-=-. It is very desirable to obtain such an algorithm when the combinatorial class to be sampled displays simple enumerative properties, like Formula (1) for triangulations. New results. The central resu... |

13 |
A bijection for triangulations of a polygon with interior points and multiple edges. Theoret
- Poulalhon, Schaeffer
(Show Context)
Citation Context ...ole played in this paper by minimal realizers of triangulations is akin to the role of breadth-first search spanning trees in planar maps [28], and of minimal bipolar orientations in 2-connected maps =-=[24]-=-. Our bijection allows us to address the three previously discussed problems. From the coding point of view, our encoding in terms of trees preserves the� entropy and satisfies linearity: each triangu... |

11 | Planar maps and Airy phenomena
- Banderier, Flajolet, et al.
(Show Context)
Citation Context ...maps, 4-regular maps). It later turned out that constraints of this kind lead systematically to explicit enumeration results for subclasses of maps (in the form of algebraic generating functions, see =-=[5]-=- and references therein). A natural question in this context is to find simple combinatorial proofs explaining these results, as opposed to the technical computational proofs à la Tutte. This was done... |

9 |
A linear-time algorithm for the generation of trees. Algorithmica 17
- Alonso, Remy, et al.
- 1997
(Show Context)
Citation Context ...rm distribution [15]. As a result these perfect samplers often operate in linear time with little more than the amount of random bits required by information theory bounds to generate a configuration =-=[2, 13]-=-. It is very desirable to obtain such an algorithm when the combinatorial class to be sampled displays simple enumerative properties, like Formula (1) for triangulations. New results. The central resu... |

8 |
Wagner’s theorem on realizers
- Bonichon, Saëc, et al.
(Show Context)
Citation Context |

6 |
Effective sampling of random surfaces by baby universe surgery, Phys
- Ambjørn, Bialas, et al.
- 1994
(Show Context)
Citation Context ...btained by sampling from the uniform distribution on rooted triangulations [4]. Several approximate sampling algorithms were thus developed by physicists for planar maps, including for triangulations =-=[3]-=-. Most of them are based on Markov chains, the mixing times of which are not known (see however [17] for a related study). A recursive perfect sampler was also developed for cubic maps, but has at lea... |

3 |
Geometry of a two-dimensional quantum gravity: numerical study
- Agishtein, Migdal
- 1991
(Show Context)
Citation Context ... on Markov chains, the mixing times of which are not known (see however [17] for a related study). A recursive perfect sampler was also developed for cubic maps, but has at least quadratic complexity =-=[1]-=-. More efficient and perfect samplers were recently developed for a dozen of classes of planar maps [5, 28]. These algorithms are linear for triangular maps (with multiple edges allowed) but have aver... |

3 |
Fraysseix and P. Ossona de Mendez. On Regular Orientations
- de
- 1994
(Show Context)
Citation Context ...lors. v0 v2 7s8 c0 c1 c2 Fig. 7. Local property of a realizer. From now on, this second condition is referred to as Schnyder condition. Realizers of triangulations satisfy a number of nice properties =-=[12, 14, 29]-=-, among which we shall use the following ones: Proposition 1. – Every triangulation has a realizer. – The set of realizers of a triangulation can be endowed with an order for which there is a unique m... |

3 |
Enumeration of rooted planar triangulations with respect to diagonal flips, J Combin Theory Ser A 88(2
- Gao, Wang
- 1999
(Show Context)
Citation Context ... sampling algorithms were thus developed by physicists for planar maps, including for triangulations [3]. Most of them are based on Markov chains, the mixing times of which are not known (see however =-=[17]-=- for a related study). A recursive perfect sampler was also developed for cubic maps, but has at least quadratic complexity [1]. More efficient and perfect samplers were recently developed for a dozen... |